Ab-initio shell model with a core

We construct effective two- and three-body Hamiltonians for the $p$-shell by performing $12\hslash \Omega$ ab initio no-core shell model (NCSM) calculations for $A=6$ and 7 nuclei and explicitly projecting the many-body Hamiltonians onto the $0\hslash \Omega$ space. We then separate these effective Hamiltonians into inert core, one- and two-body contributions (also three-body for $A=7$) and analyze the systematic behavior of these different parts as a function of the mass number $A$ and size of the NCSM basis space. The role of effective three- and higher-body interactions for $A>6$ is investigated and discussed.

Figure 1

The excitation energies of the $J^{\pi }$ states and ground state energy for ${}^6\mathrm{Li}$ calculated in the $N_{\max }\hslash \Omega$ spaces with the INOY interaction and $\hslash \Omega =14$ MeV. The experimental spectra and ground state energy are shown for comparison.

Figure 2

The diagonal TBMEs of the sum for the core and one-body parts, $\langle \mathit{ab}|H_0^{6,4}+H_1^{6,5}|\mathit{ab}\rangle$, for the effective Hamiltonian, ${\mathcal{H}}_{6,6}^{0,N_{\max }}$, for ${}^6\mathrm{Li}$ as a function of $N_{\max }$. The corresponding curves are labeled by quantum numbers $2j_{a}2j_{b}2j_{a}2j_b$.

Figure 3

The diagonal $\mathit{pn}$ TBMEs of the two-body part, $\langle \mathit{ab}\left|V_2^{6,6}\right|\mathit{ab}\rangle {}_{\mathit{JT}}$, of the effective Hamiltonian, ${\mathcal{H}}_{6,6}^{0,N_{\max }}$, as a function of $N_{\max }$. The corresponding curves are labeled by quantum numbers $2j_{a}2j_{b}2j_{a}2j_b$ and JT.

Figure 4

The nondiagonal $\mathit{pn}$ TBMEs of the two-body part, $\langle \mathit{ab}\left|V_2^{6,6}\right|\mathit{cd}\rangle {}_{\mathit{JT}}$, for the effective Hamiltonian, ${\mathcal{H}}_{6,6}^{0,N_{\max }}$, as a function of $N_{\max }$. The corresponding curves are labeled by quantum numbers $2j_{a}2j_{b}2j_{c}2j_d$ and JT.

Figure 5

The ground state energy, $E_{\mathrm{g}.\mathrm{s}.}$, of ${}^7\mathrm{Li}$ as a function of $N_{\max }$. The NCSM results with the $H_{A=7,2}^{N_{\max },\Omega }$ Hamiltonian are shown by the filled circles connected with the solid line. The SSM results with the effective ${\mathcal{H}}_{7,6}^{0,N_{\max }}$ Hamiltonian decomposed according to Eq. (18) are shown by squares connected with the dashed line. The SSM results with the effective ${\mathcal{H}}_{7,6}^{0,N_{\max }}$ Hamiltonian decomposed according to Eq. (19) are shown by the filled circles connected with a dashed line.

Figure 6

NCSM (solid line) and SSM [using Eq. (18), dashed line] spectra for ${}^7\mathrm{Li}$. The states with spin $J$ are marked by $2J$.

Figure 7

The difference of the NCSM and SSM [Eq. (20)] ground state energies for different values of mass number $A$ as a function of isospin projection $T_z=(N-Z)/2$.

Figure 8

Comparison of spectra for ${}^8\mathrm{He}$, ${}^9\mathrm{He}$, and ${}^{10}\mathrm{He}$ from SSM calculations using the effective 2BVC and 3BVC Hamiltonians and from exact NCSM calculation for $N_{\max }=6$ and $\hslash \Omega =14$ MeV using the INOY interaction.

Figure 9

Comparison of spectra for ${}^8\mathrm{He}$, ${}^9\mathrm{He}$, and ${}^{10}\mathrm{He}$ from SSM calculations using the effective 2BVC and 3BVC Hamiltonians and from exact NCSM calculation for $N_{\max }=6$ and $\hslash \Omega =20$ MeV using the CD-Bonn interaction.